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Point Cloud Synthesis Using Inner Product Transforms
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Proposes the Inner Product Transform (IPT), an injective, efficient 2D image encoding of point clouds via inner products with sampled directions, and generates point clouds by generating IPT images with a VAE and decoding them with a CNN IP-Encoder.
Problem
Point cloud synthesis is hard due to sparsity and set structure; state-of-the-art generative models are slow to train and sample. The paper asks whether a novel point cloud representation can give large computational gains without sacrificing quality.
Approach
Represent a point cloud as a 2D image (Inner Product Transform) counting points below sampled hyperplanes, with a sigmoid relaxation that is differentiable; generate that image with any image generative model (here a CNN VAE) and reconstruct the point cloud with a lightweight 1D-convolutional IP-Encoder. Proves injectivity with n+1 affinely independent directions and metric/linearity/surjectivity
Key results
- IP-Encoder reconstruction gives airplane EMD 1.57 vs PointFlow 5.36 and Canonical VAE 3.19, ranking second behind LION on EMD; training about 30 min vs 192 h for PointFlow.
- IP-VAE achieves best MMD-CD and MMD-EMD on airplane (0.1901 / 0.3383) and chair/car MMD-EMD among compared models, but weaker COV and 1-NNA.
- IP-VAE inference takes 0.00079 s on GPU versus 0.27 s (PointFlow) and 29.9 s (PVD), orders of magnitude faster.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| combine CD and IPT-64 losses rather than use CD loss alone (ablation) | IP-Encoder · ShapeNet reconstruction | EMD: 8.89 \pm 0.0 -> 1.46 \pm 0.0 (Airplane); 32.43 \pm 0.0 -> 8.44 \pm 0.0 (Chair); 14.96 \pm 0.0 -> 4.16 \pm 0.0 (Car) | Helped |
| combine CD and IPT-64 losses rather than use IPT-64 loss alone (ablation) | IP-Encoder · ShapeNet reconstruction | CD: 2.41 \pm 0.0 -> 1.03 \pm 0.0 (Airplane); 13.06 \pm 0.0 -> 9.52 \pm 0.0 (Chair); 7.75 \pm 0.0 -> 6.12 \pm 0.0 (Car) | Helped |
| combine CD and IPT-64 losses rather than use IPT-64 loss alone (ablation) | IP-Encoder · ShapeNet reconstruction | EMD: 1.09 \pm 0.0 -> 1.46 \pm 0.0 (Airplane); 4.29 \pm 0.0 -> 8.44 \pm 0.0 (Chair); 2.47 \pm 0.0 -> 4.16 \pm 0.0 (Car) | Mixed |
| increase IPT direction count from 4 to 128 (ablation) | IP-Encoder · ShapeNet Airplane reconstruction | CD: 1.287 -> 1.05 | Helped |
| increase IPT direction count from 4 to 128 (ablation) | IP-Encoder · ShapeNet Airplane reconstruction | EMD: 2.047 -> 1.57 | Helped |
| use scale 32 instead of scale 128 at resolution 128 (ablation) | IPT optimization · ShapeNet Airplane inversion | CD: 1.27 \pm 0.0 -> 0.52 \pm 0.0 | Helped |
| use scale 32 instead of scale 128 at resolution 128 (ablation) | IPT optimization · ShapeNet Airplane inversion | EMD: 0.29 \pm 0.0 -> 0.12 \pm 0.0 | Helped |
| use IP-Downsampler instead of uniform subsampling | ShapeNet point-cloud downsampling and reconstruction · ShapeNet Airplane | CD: 3.10 -> 1.18 | Helped |
| use IP-Downsampler instead of uniform subsampling | ShapeNet point-cloud downsampling and reconstruction · ShapeNet Airplane | EMD: 4.71 -> 2.58 | Helped |
| replace IP-Encoder with backpropagation optimization for clean IPT inversion | ShapeNet reconstruction · ShapeNet Airplane | CD: 1.05 \pm 0.0 -> 0.52 \pm 0.0 | Helped |
| increase IPT resolution from 128 to 64 (ablation) | IPT optimization · ShapeNet reconstruction | CD: 0.52 \pm 0.0 -> 1.22 \pm 0.0 (Airplane, best scale at each resolution) | Hurt |
| use backpropagation optimization on generated or approximate IPTs | IPT inversion · Generated IPT reconstruction | reconstruction quality | Hurt |
Surprising findings
- Backpropagation directly through the approximate IPT outperforms the learned IP-Encoder in reconstruction quality, but fails on approximate (generated) IPTs.
- The IP-Encoder reconstructs partial-view point clouds and interpolated IPTs without retraining, despite being trained only on complete shapes.
Limitations
- Not invariant to rotations; relies on data augmentation to learn approximate equivariance.
- Generative diversity (COV, 1-NNA) is only middling because the simple VAE generator is weak.
- Sampling directions in high dimensions is inefficient, and injectivity does not guarantee finding suitable directions.
Experimental setup
Datasets: ShapeNet MNIST ShapeNet13 GenRe
Benchmarks: ShapeNet
Metrics: Chamfer Distance Earth Mover's Distance MMD COV 1-NNA inference time
Baselines: PointFlow SoftFlow ShapeGF Canonical VAE LION SetVAE Point Voxel Diffusion Point Straight Flows XCube
Builds on: Euler Characteristic Transform beta-VAE
Closest papers by meaning
- Learning Progressive Point Embeddings for 3D Point Cloud Generation (CVPR 2021 · similarity 0.86)
- Density-Preserving Deep Point Cloud Compression (CVPR 2022 · similarity 0.85)
- Generative PointNet: Deep Energy-Based Learning on Unordered Point Sets for 3D Generation, Reconstruction and Classification (CVPR 2021 · similarity 0.85)
- Point-BERT: Pre-Training 3D Point Cloud Transformers With Masked Point Modeling (CVPR 2022 · similarity 0.85)
- PointInfinity: Resolution-Invariant Point Diffusion Models (CVPR 2024 · similarity 0.84)
- Dataset Distillation of 3D Point Clouds via Distribution Matching (NeurIPS 2025 · similarity 0.84)
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